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Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 RationalInterpolation

Source Notebook

Find a rational interpolation of a function

Contributed by: Wolfram Research

ResourceFunction["RationalInterpolation"][expr,{x,m,n},{x1,x2,…,xn+m+1}]

gives the rational interpolant to expr with numerator order m and denominator order n, where x1,x2,… are the abscissas of the interpolation points.

ResourceFunction["RationalInterpolation"][expr,{x,m,n},{x,x0,x1}]

gives the rational interpolant with the interpolation points chosen automatically on the interval x0 to x1.

Details and Options

The following options can be given:
WorkingPrecisionMachinePrecisionprecision to use in internal computations
"Bias"0bias in the automatic choice of interpolation points
Values for the "Bias" option must be numbers between -1 and 1. A positive value causes the points to be biased to the right and zero causes the points to be distributed symmetrically.

Examples

Basic Examples (2) 

A rational interpolation of degree (2,4) to ex at seven equally spaced points between 0 and 2:

In[1]:=
ResourceFunction["RationalInterpolation"][E^x, {x, 2, 4}, Range[0, 2, 1/3]]
Out[1]=

The error between the function and the approximation tends to get larger near the endpoints:

In[2]:=
Plot[% - E^x, {x, 0, 2}, PlotRange -> All]
Out[2]=

Automatically choosing the interpolation points results in a smaller maximum error:

In[3]:=
ResourceFunction["RationalInterpolation"][E^x, {x, 2, 4}, {x, 0, 2}]
Out[3]=
In[4]:=
Plot[% - E^x, {x, 0, 2}]
Out[4]=

Options (1) 

Bias (1) 

Bias the distribution of the points to the right to get smaller errors there and larger errors to the left:

In[5]:=
ResourceFunction["RationalInterpolation"][E^x, {x, 2, 4}, {x, 0, 2}, "Bias" -> .25]
Out[5]=
In[6]:=
Plot[% - E^x, {x, 0, 2}]
Out[6]=

Requirements

Wolfram Language 11.3 (March 2018) or above

Version History

  • 1.0.0 – 29 March 2019

Related Resources

License Information